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Under the Strong Exponential Time Hypothesis, an integer linear program with $n$ Boolean-valued variables and $m$ equations cannot be solved in $c^n$ time for any constant $c < 2$. If the domain of the variables is relaxed to $[0,1]$, the associated linear program can of course be solved in polynomial time. In this work, we give a natural algorithmic bridging between these extremes of $0$-$1$ and linear programming. Specifically, for any subset (finite union of intervals) $E subset [0,1]$ containing ${0,1}$, we give a random-walk based algorithm with runtime $O_E((2-text{measure}(E))^ntext{poly}(n,m))$ that finds a solution in $E^n$ to any $n$-variable linear program with $m$ constraints that is feasible over ${0,1}^n$. Note that as $E$ expands from ${0,1}$ to $[0,1]$, the runtime improves smoothly from $2^n$ to polynomial. Taking $E = [0,1/k) cup (1-1/k,1]$ in our result yields as a corollary a randomized $(2-2/k)^{n}text{poly}(n)$ time algorithm for $k$-SAT. While our approach has some high level resemblance to Sch{o}nings beautiful algorithm, our general algorithm is based on a more sophisticated random walk that incorporates several new ingredients, such as a multiplicative potential to measure progress, a judicious choice of starting distribution, and a time varying distribution for the evolution of the random walk that is itself computed via an LP at each step (a solution to which is guaranteed based on the minimax theorem). Plugging the LP algorithm into our earlier polymorphic framework yields fast exponential algorithms for any CSP (like $k$-SAT, $1$-in-$3$-SAT, NAE $k$-SAT) that admit so-called `threshold partial polymorphisms.
We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by provid
The celebrated multi-armed bandit problem in decision theory models the basic trade-off between exploration, or learning about the state of a system, and exploitation, or utilizing the system. In this paper we study the variant of the multi-armed ban
For a graph $G$ on $n$ vertices, naively sampling the position of a random walk of at time $t$ requires work $Omega(t)$. We desire local access algorithms supporting $text{position}(G,s,t)$ queries, which return the position of a random walk from som
We introduce the combinatorial optimization problem Time Disjoint Walks (TDW), which has applications in collision-free routing of discrete objects (e.g., autonomous vehicles) over a network. This problem takes as input a digraph $G$ with positive in
Since the elimination algorithm of Fourier and Motzkin, many different methods have been developed for solving linear programs. When analyzing the time complexity of LP algorithms, it is typically either assumed that calculations are performed exactl